New Techniques for Approximating Optimal Substructure Problems in
Power-Law Graphs
Yilin Shen, Dung T. Nguyen, Ying Xuan, My T. Thai
Department of Computer Information Science and Engineering
University of Florida, Gainesville, FL, 32611
{yshen, dtnguyen, yxuan, mythai}@cise.ufl.edu
Abstract
The remarkable discovery of many large-scale real networks is the power-law distribution in degree
sequence: the number of vertices with degree i is proportional to i−β for some constant β > 1.
A lot of researchers believe that it may be easier to solve some optimization problems in powerlaw graphs. Unfortunately, many problems have been proved NP-hard even in power-law graphs.
Intuitively, a theoretical question is raised: Are these problems on power-law graphs still as hard
as on general graphs?
In this paper, we show that many optimal substructure problems, such as Minimum Dominating Set, Minimum Vertex Cover and Maximum Independent Set, are easier to solve in
power-law graphs by illustrating better inapproximability factors. An optimization problem has the
property of optimal substructure if its optimal solution on some given graph is essentially the union
of the optimal sub-solutions on all maximal connected components. In particular, we prove the
above problems and a more general problem (ρ-Minimum Dominating Set) remain APX-hard
and their constant inapproximability factors on general power-law graphs by using the cycle-based
embedding technique to embed any d-bounded graphs into a power-law graph. In addition, in
simple power-law graphs, we further prove the corresponding inapproximability factors of these
problems based on the graphic embedding technique as well as that of Maximum Clique and
Minimum Coloring using the embedding technique in [1]. As a result of these inapproximability
factors, the belief that there exists some (1 + o(1))-approximation algorithm for these problems
on power-law graphs is proven not always true. At last, we do the in-depth investigations in the
relationship between the exponential factor β and constant greedy approximation algorithms.
Keywords: Power-Law Graphs, Optimal Substructure Problems, Embedding Techniques,
Inapproximability Optimal Substructure Framework, Complexity Hardness, Inapproximability
1. Introduction and Related Work
A great number of large-scale networks in real life are discovered to follow a power-law distribution in their degree sequences, ranging from the Internet [2], the World-Wide Web (WWW)
[3] to social networks [4]. That is, the number of vertices with degree i is proportional to i−β for
some constant β in these graphs, which is called power-law graphs. The observations show that
the exponential factor β ranges between 1 and 4 for most real-world networks [5]. Intuitively, the
following theoretical question is raised: What are the differences in terms of complexity hardness
Preprint submitted to Theoretical Computer Science
May 4, 2011
and inapproximability factor of several optimization problems between in general graphs and in
power-law graphs?
Many experimental results on random power-law graphs give us a belief that the problems
might be much easier to solve on power-law graphs. Eubank et al. [6] showed that a simple
greedy algorithm leads to a 1 + o(1) approximation factor on Minimum Dominating Set (MDS)
and Minimum Vertex Cover (MVC) on power-law graphs (without any formal proof) although
MDS and MVC has been proved NP-hard to be approximated within (1 − ) log n and 1.366 on
general graphs respectively [7]. In [8], Gopal also claimed that there exists a polynomial time
algorithm that guarantees a 1 + o(1) approximation of the MVC problem with probability at least
1 − o(1). Unfortunately, there is no such formal proof for this claim either. Furthermore, several
papers also have some theoretical guarantees for some problems on power-law graphs. Gkantsidis
et al. [9] proved the flow through each link is at most O(n log2 n) on power-law random graphs
where the routing of O(du dv ) units of flow between each pair of vertices u and v with degrees
du and dv . In [9], the authors take advantage of the property of power-law distribution by using
the structural random model [10, 11] and show the theoretical upper bound with high probability
1 − o(1) and the corresponding experimental results. Likewise, Janson et al. [12] gave an algorithm
that approximated Maximum Clique within 1 − o(1) on power-law graphs with high probability
on the random poisson model G(n, α) (i.e. the number of vertices with degree at least i decreases
roughly as n−i ). Although these results were based on experiments and various random models,
they raise an interest in investigating hardness and inapproximability of optimization problems on
power-law graphs.
Recently, Ferrante et al. [1] had an initial attempt on power-law graphs to show the NPhardness of Maximum Clique (Clique) and Minimum Graph Coloring (Coloring) (β > 1)
by constructing a bipartite graph to embed a general graph into a power-law graph and NPhardness of MVC, MDS and Maximum Independent Set (MIS) (β > 0) based on their optimal
substructure properties. Unfortunately, there is a minor flaw in the proof of their Lemma 5 which
makes the proof of NP-hardness of MIS, MVC, MDS with β < 1 no longer hold. Then we present
another way in APPENDIX A to show the NP-hardness of these problems when β < 1 so as to fix
this non-trivial flaw.
Our Contributions: In this paper, we propose two new techniques on optimal substructure
problems, Cycle-Based Embedding Technique and Graphic Embedding Technique, to embed a dbounded graph into a general power-law graph and a simple power-law graph respectively. Then
we use these two techniques to further prove the APX-hardness and the inapproximability of MIS,
MDS, and MVC on general power-law graphs and simple power-law graphs. These inapproximability results on power-law graphs are shown in Table 1. Furthermore, the inapproximability results
in Clique and Coloring are shown by taking advantage of the reduction in [1]. We also analyze
the relationship between β and constant greedy approximation algorithms for MIS and MDS.
In addition, due to a lot of recent studies in online social networks on the influence propagation
problem [13, 14], we formulate this problem as ρ-Minimum Dominating Set (ρ-MDS) and show it
hard to be approximated within 2 − (2 + od (1)) log log d/ log d factor on d-bounded graphs under
unique games conjecture, which further leads to the following inapproximability result on power-law
graphs (shown in Table 1).
The rest of paper is organized as follows. In Section 2, we introduce some problem definitions, the model of power-law graphs, and some related concepts. The inapproximability optimal
substructure framework is presented in Section 3. We show the hardness and inapproximability
2
Table 1: Inapproximability Factors on Power-Law Graphs with Exponential Factor β > 1
Problem
MIS
MDS
MVC, ρ-MDS
Clique
Coloring
General Power-Law Graph
1
1 + 140(2ζ(β)3
β −1) − ε
1
1 + 390(2ζ(β)3β −1)
1+
log c
2 1−(2+oc (1)) log
log c
1
ζ(β)cβ +c β (c+1)
Simple Power-Law Graph
1
1 + 1120ζ(β)3
β − ε
1
1 + 3120ζ(β)3β
1+
log c
2−(2+oc (1)) log
log c
2ζ(β)cβ (c+1)
O n1/(β+1)−
O n1/(β+1)−
-
a
Conditions: MIS and MDS: P6=NP; MVC, ρ-MDS: unique games conjecture; Clique, Coloring:
NP6=ZPP.
b c is a constant which is the smallest d satisfying the condition in [15].
of MIS, MDS, MVC in general power-law graphs using the cycle-based embedding technique in
Section 4. More inapproximability results in simple power-law graphs are illustrated in Section 5
based on the graphic embedding technique, which implies the APX-hardness of these problems.
Additionally, the inapproximability factor on maximum clique and minimum coloring problems
are proven. In Section 6, we analyze the relationship between β and constant approximation algorithms, which further proves that the integral gap is typically small for optimization problems
on power-law graphs than that on general bounded graphs. In Appendix A, we fix the flaw in the
NP-hardness proof for β < 1 presented in [1].
2. Preliminaries
In this section, we first recall the definition of several classical optimization problems and
formulate the new optimization problem ρ-Minimum Dominating Set. Then the power-law model
and some corresponding concepts are proposed. At last, we introduce some special graphs which
will be used in the analysis throughout the whole paper.
2.1. Problem Definitions
Definition 2.1 (Maximum Independent Set). Given an undirected graph G = (V, E), find a
subset S ⊆ V with the maximum size such that no two vertices in S are adjacent.
Definition 2.2 (Minimum Vertex Cover). Given an undirected graph G = (V, E), find a subset S ⊆ V with the minimum size such that for each edge E at least one endpoint belongs to
S.
Definition 2.3 (Minimum Dominating Set). Given an undirected graph G = (V, E), find a
subset S ⊆ V with the minimum size such that for each vertex vi ∈ V \ S, at least one neighbor of
vi belongs to S.
Definition 2.4 (Maximum Clique). Given an undirected graph G = (V, E), find a clique with
maximum size where a subgraph of G is called a clique if all its vertices are pairwise adjacent.
Definition 2.5 (Minimum Graph Coloring). Given an undirected graph G = (V, E), label
the vertices in V with minimum number of colors such that no two adjacent vertices share the
same color.
3
The ρ-Minimum Dominating Set is defined as general version of MDS problem. In the context
of influence propagation, the ρ-MDS problem aims to find a subset of nodes with minimum size
such that all nodes in the whole network can be influenced within t rounds. In particular, a node is
influenced when ρ fraction of its neighbors are influenced. For simplicity, we define ρ-MDS problem
in the case that t = 1.
Definition 2.6 (ρ-Minimum Dominating Set). Given an undirected graph G = (V, E), find a
subset S ⊆ V with the minimum size such that for each vertex vi ∈ V \ S, |S ∩ N (vi )| ≥ ρ|N (vi )|.
2.2. Power-Law Model and Some Notations
A great number of models [16, 17, 10, 11, 18] on power-law graphs are emerging in the past
recent years. In this paper, we do the analysis based on the general (α, β) model, that is, the
graphs only constrained by the power-law distribution in degree sequences. We first define the
following two types of degree sequences.
Definition 2.7 (y-Degree Sequence). Given a graph G = (V, E), the y-degree sequence of G is
a sequence Y = hy1 , y2 , . . . , y∆ i where ∆ is the maximum degree of G and yi = |{u|u ∈ V ∧deg(u) =
i}|.
Definition 2.8 (d-Degree Sequence). Given a graph G = (V, E), the d-degree sequence of G is
a sequence D = hd1 , d2 , . . . , dn i of vertex in non-increasing order of their degrees.
Note that y-degree sequence and d-degree sequence are interchangeable. Given a y-degree
sequence Y = hy1 , y2 , . . . , y∆ i, the corresponding d-degree sequence is D = h∆, ∆, . . . , ∆ − 1, ∆ −
1, . . . , ∆ − 1, . . . , 1, . . . , 1i where the number i appears yi times. Because of their equivalence, we
may use only y-degree sequence or d-degree sequence or both without changing the meaning or
validity of results. The definition of power-law graphs can be expressed via y-degree sequences as
follows.
Definition 2.9 (General (α, β) Power-Law Graph Model). A graph G = (V, E) is called a
(α, β) power-law graph G(α,β) where multi-edges and self-loops are allowed if the maximum degree
is ∆ = eα/β and the number of vertices of degree i is:
(
P α β
is even
eα /iβ ,
if i > 1 or ∆
i=1 e /i
(2.1)
yi =
α
be c + 1,
otherwise
In simple (α, β) power-law graphs, there are no multi-edges and self-loops.
Note that a power-law graph are represented by two parameters α and β. Since graphs with
the same β express the same behaviors, we categorize all graphs with the same β into a β-family
of graphs such that β is regarded as a constant instead of an input. In addition, we only consider
the case β > 1 because almost all real large-scale networks have β > 1. In this case, the number
of vertices is:
∆
X
1
eα
= ζ(β)eα + n β ≈ ζ(β)eα
β
i
i=1
P
1
where ζ(β) = ∞
i=1 iβ is the Riemann Zeta Function. Also the d-degree sequence of any (α, β)
power-law graph is continuous according to the following definition.
4
u1
u2
Definition 2.10 (Continuous Sequence). An integer sequence hd1 , d2 , . . . , dn i, where d1 ≥ d2 ≥
· · · ≥ dn , is continuous if ∀1 ≤ i ≤ n − 1, |di − di+1 | ≤ 1.
v1
v1
v2
v2
v3if there exists a graph
Definition 2.11 (Graphic Sequence). uA8 sequence D is said to be graphic
v8
v8
v3
such that D is its d-degree sequence.
v3
Definition 2.12 (Degree Set). Given a graph G, let Di (G) be the set of vertices of degree i on
v7
v7
v4
G.
v4
u7
Furthermore, we define the d-bounded graph as
v6
u4
v5
v6
v5
Definition 2.13 (d-Bounded Graph). Given a graph
u6 G = (V,
u5E), G is a d-bounded graph if
the degree of any vertex is upper bounded by an integer constant d.
2.3. Special Graphs
u1
u2
v1
u8
u5
v2
v8
v3
v7
v4
u7
v6
v5
v3
u4
u4
v1
u3
v2
u2
u6
v4
u6
u1
u5
v3
(b) ~2 − C82
(a) RC83
Figure 1: Special Graph Examples: The left one is a (3, 3, 3, 3, 3, 3, 3, 3)-regular cycle and the right one is a (3, 3, 3, 3)branch-(2, 2, 2, 2, 2, 2)-cycle. The grey vertices consist of the optimal solution of MDS on these two special graphs.
~
~
~
cycle
Definition 2.14 (d-Regular
Cycle RCnd ). Given a vector d~ = (d1 , . . . , dn ), a d-regular
d
th
RCn is composed of two cycles. Each cycle has n vertices and two i vertices in each cycle are
~
adjacent with each other by di − 2 multi-edges. That is, d-regular
cycle RCnd has 2n vertices and
the two ith vertex has the same degree di . An example RC8d is shown in Figure 1(a).
~
~
Definition 2.15 (~κ-Branch-d-Cycle
~κ-BCnd ). Given two vectors d~ = (d1 , . . . , dn ) and ~κ = (κ1 , . . . , κm ),
~
the ~κ-branch-d-cycle
is composed of a cycle with a number of vertices n such that each vertex has
degree di as well as |~κ|/2 appendant branches, where |κ| is a even number. Note that any ~κ~
branch-d-cycle
has |~κ| even number of vertices with odd degrees. An example is shown in Figure
1(b).
5
2.4. Existing Inapproximability Results
Here we list some inapproximability results in the literature to use later in our proofs.
(1) In d-bounded graphs, MVC is hard to be approximated into 2 − (2 + od (1)) log log d/ log d for
every sufficiently large integer d under unique games conjecture [15, 19].
140
(2) In 3-bounded graphs, MIS and MDS is NP-hard to be approximated into 139
− ε for any
391
ε > 0 and 390 respectively [20].
(3) Maximum clique and minimum coloring problem is hard to be approximated into n1− on
general graphs unless NP=ZPP [21].
3. Inapproximability Optimal Substructure Framework in Power-Law Graphs
In this section, we introduce a framework to derive the approximation hardness of optimal substructure problems on power-law graphs. A graph optimization problem is said to satisfy optimal
substructure if its optimal solution is the union of the optimal solutions on each connected component. Therefore, when a graph G is embedded into a power-law graph G0 , the optimal solution
in G0 consists of a subset of the optimal solution in G. According to this important property, we
present the Inapproximability Optimal Substructure Framework to prove the inapproximability factor if there exists a Embedded-Approximation-Preserving Reduction that relates the approximation
hardness in general graphs and power-law graphs by guaranteeing the relationship between the
solutions in the original graph and the constructed graph.
Definition 3.1 (Embedded-Approximation-Preserving Reduction). Given an optimal substructure problem O, a reduction from an instance on graph G = (V, E) to another instance on
a power-law graph G0 = (V 0 , E 0 ) is called embedded-approximation-preserving if it satisfies the
following properties:
(1) G is a subset of maximal connected components of G0 ;
(2) The optimal solution of O on G0 , OP T (G0 ), is upper bounded by COP T (G) where C is a
constant correspondent to the growth of the optimal solution.
Theorem 3.1 (Inapproximability Optimal Substructure Framework). Given an optimal
substructure problem O, if there exists an embedded-approximation-preserving reduction from a
graph G to another graph G0 , we can extract the inapproximability factor δ of O on G0 using
C
-inapproximability of O on G, where δ is lower bounded by (C−1)+1
and +C−1
when O is a
C
maximum and minimum optimization problem respectively.
Proof. Suppose that there exists an algorithm providing a solution of O on G0 with size at most
δ times the optimal solution. Denote A and B to be the sizes of the produced solution on G and
G0 \ G and A∗ and B ∗ to be their corresponding optimal values. Hence, we have B ∗ ≤ (C − 1)A∗ .
With the completeness that OP T (G) = A∗ ⇒ OP T (G0 ) = B ∗ , the soundness leads to the lower
bound of δ which is dependent on the type of O, maximization or minimization problem, as follows.
Case 1: When O is a maximization problem, we start from the definition of soundness as
A∗ + B ∗ ≤ δ(A + B)
∗
⇔ A ≤ δA + (δ − 1)B
∗
(3.1)
∗
⇔ A ≤ δA + (δ − 1)(C − 1)A
6
(3.2)
∗
(3.3)
where (3.2) holds since B ≤ B ∗ and (3.3) holds since B ∗ ≤ (C − 1)A∗ .
On the other hand, it is hard to approximate O within on G, thus A∗ > A. Replace it to
the above inequality, we have:
A∗ < A∗ δ/ + (δ − 1)(C − 1)A∗ ⇔ δ >
C
(C − 1) + 1
Case 2: When O is a minimization problem, since B ∗ ≤ B, similarly
A + B ≤ δ(A∗ + B ∗ )
⇔ A ≤ δA∗ + (δ − 1)B ∗
⇔ A ≤ δA∗ + (δ − 1)(C − 1)A∗
Then from A > A∗ ,
< δ + (δ − 1)(C − 1) ⇔ δ >
+C−1
C
4. Hardness and Inapproximability of Optimal Substructure Problems on General
Power-Law Graphs
4.1. General Cycle-Based Embedding Technique
In this section, we propose a General Cycle-Based Embedding Technique on (α, β) power-law
graphs with β > 1. The basic idea is to embed an arbitrary d-bounded graph into power-law
graphs using a d~1 -regular cycle, a ~κ-branch-d~2 -cycle and a number of cliques K2 , where d~1 , d~2 and
~κ are defined by α and β. Before discussing the main embedding technique, we first show that most
~
~
optimal substructure problems can be polynomially solved in both d-regular
cycles and ~κ-branch-dcycle. In this context, the cycle-based embedding technique helps to prove the complexity of these
optimal substructure problems on power-law graphs according to their corresponding complexity
results on general bounded graphs.
~
Lemma 4.1. MDS, MVC and MIS are polynomially solvable on d-regular
cycles.
~
Proof. Here we just prove MDS problem is polynomially solvable on d-regular
cycles. The algorithm is simple. From an arbitrarily vertex, we select the vertex on the other cycle in two hops.
The algorithm will terminate until all vertices are dominated. Now we will show that this gives
the optimal solution. Let’s take RC83 as an example. As shown in Figure 1(a), the size of MDS is
4. Notice that each vertex can dominate exact 3 vertices, that is, 4 vertices can dominate exactly
12 vertices. However, in RC83 , there are altogether 16 vertices, which have to be dominated by at
least 4 vertices apart from the vertices in MDS. That is, the algorithm returns an optimal solution.
The proof of MVC and MIS is similar.
~
Lemma 4.2. MDS, MVC and MIS is polynomially solvable on ~κ-branch-d-cycles.
Proof. Again we show the proof of MDS. First we select the vertices connecting both the branches
and the cycle. Then by removing the branches, we will have a line graph regardless of self-loops,
on which MDS is polynomially solvable. It is easy to see that the size of MDS will increase if any
one vertex connecting both the branch and the cycle in MDS is replaced by some other vertices.
The proof of MIS is similar. Note that the optimal solution for MVC consists of all vertices since
all edges need to be covered.
7
Theorem 4.1 (Cycle-Based Embedding Technique). Any d-bounded graph Gd can be embedded into a power-law graph G(α,β) with β > 1 such that Gd is a maximal component and most
optimal substructure problems can be polynomially solvable on G(α,β) \ Gd .
Proof. With the given β, we choose α to be max{ln max1≤i≤d {ni · iβ }, β ln d}. Based on τ (i) =
beα /iβ c − ni where ni = 0 when i > d, we construct the power-law graph G(α,β) as the following
Algorithm 1. The last step holds since the number of vertices of odd degrees has to be even. From
Step 1, we know eα = max{max1≤i≤d {ni · iβ }, dβ } ≤ dβ n, that is, the number of vertices N in
graph G(α,β) satisfies N ≤ ζ(β)dβ n, which means that N/n is a constant. According to Lemma
4.1 and Lemma 4.2, since G(α,β) \ Gd is composed of a d~1 -regular cycle and a d~1 -branch-d~2 -cycle,
2
it can be polynomially solvable. Note that the number of vertices in L is at most ∆ since there is
at most one leftover vertex of each degree.
Algorithm 1: Cycle Embedding Algorithm
1
2
3
4
5
α ← max{ln max1≤i≤d {ni · iβ }, β ln d};
For τ (1) vertices of degree 1, add bτ (1)/2c number of cliques K2 ;
For τ (2) vertices of degree 2, add a cycle with the size τ (2);
For all vertices of degree larger than 2 and smaller than ∆, construct a d~1 -regular cycle
where d~1 is a vector composed of bτ (i)/2c number of elements i for all i satisfying τ (i) > 0;
For all leftover isolated vertices L such that τ (i) − 2bτ (i)/2c = 1, construct a
d~1 -branch-d~2 -cycle, where d~1 and d~2 are the vectors containing odd and even elements
2
2
2
2
correspondent to the vertices of odd and even degrees in L respectively.
4.2. APX-Hardness
In this section, we prove that MIS, MDS, MVC remain APX-hard even on power-law graphs.
Theorem 4.2. MDS is APX-hard on power-law graphs.
Proof. According to Theorem 4.1, we use the cycle-based embedding technique to show Lreduction from MDS on any d-bounded graph Gd to MDS on a power-law graph G(α,β) since
MDS is proven APX-hard on d-bounded graphs [22].
Letting φ be a feasible solution on Gd , we can construct MDS in G0 such that MDS on a K2 is 1,
~
~
n/4 on a d-regular
cycle and n/3 on a cycle and a ~κ-branch-d-cycle.
Therefore, for a solution φ on
Gd , we have a solution ϕ on G(α,β) to be ϕ = φ+n1 /2+n2 /3+n3 /4, where n1 , n2 and n3 corresponds
to τ (1), τ (2)∪L and all leftover vertices. Hence, we have OP T (ϕ) = OP T (φ)+n1 /2+n2 /3+n3 /4.
On one hand, for a d-bounded graph with vertices n, the optimal MDS is lower bounded by
n/(d + 1). Thus, we know
OP T (ϕ) = OP T (φ) + n1 /2 + n2 /3 + n3 /4
≤ OP T (φ) + (N − n)/2 ≤ OP T (φ) + (ζ(β)dβ − 1)n/2
h
i
≤ OP T (φ) + (ζ(β)dβ − 1)(d + 1)OP T (φ)/2 = 1 + (ζ(β)dβ − 1)(d + 1)/2 OP T (φ)
where N is the number of vertices in G(α,β) .
On the other hand, with |OP T (φ) − φ| = |OP T (ϕ) − ϕ|, we proved the L-reduction with
c1 = 1 + (ζ(β)dβ − 1)(d + 1)/2 and c2 = 1.
8
Theorem 4.3. MVC is APX-hard on power-law graphs.
Proof. In this proof, we show L-reduction from MVC on d-bounded graph Gd to MVC on powerlaw graph G(α,β) using cycle-based embedding technique.
Let φ be a feasible solution on Gd . We construct the solution ϕ ≤ φ + (N − n) since the optimal
~
~
solution of MVC is n/2 on K2 , cycle, d-regular
cycle and n on ~κ-branch-d-cycle.
Therefore, since
the optimal MVC on a d-bounded graph is lower bounded by n/(d + 1), we have
h
i
OP T (ϕ) ≤ 1 + (ζ(β)dβ − 1)(d + 1) OP T (φ)
On the other hand, with |OP T (φ) − φ| = |OP T (ϕ) − ϕ|, we proved the L-reduction with
c1 = 1 + (ζ(β)dβ − 1)(d + 1) and c2 = 1.
Corollary 4.1. MIS is APX-hard on power-law graphs.
4.3. Inapproximability Factors
In this section, we show the inapproximability factors on MIS, MVC and MDS on power-law
graphs respectively using the results in section 2.4.
Theorem 4.4. For any ε > 0, there is no 1 +
Maximum Independent Set on power-law graphs.
1
140(2ζ(β)3β −1)
− ε approximation algorithm for
Proof. In this proof, we construct the power-law graph G(α,β) based on cycle-based embedding
technique in Theorem 4.1 from d-bounded graph Gd . Let φ and ϕ be feasible solutions of MIS on Gd
~
and G(α,β) . Then OP T (ϕ) composed of OP T (φ), clique K2 , cycle, d-regular
cycle and ~κ-branch~
d-cycles
are all exactly half number of vertices. Hence, we have OP T (ϕ) = OP T (φ) + (N − n)/2
where n and N is the number of vertices in Gd and G(α,β) respectively. Since OP T (φ) ≥ n/(d + 1)
on d-bounded graphs for MIS and N ≤ ζ(β)dβ n, we further have C = 1 +
(ζ(β)dβ −1)(d+1)
2
from
N −n
(ζ(β)dβ − 1)
≤ OP T (φ) +
n
2
2
(ζ(β)dβ − 1)(d + 1)
(ζ(β)dβ − 1)(d + 1)
≤ OP T (φ) +
OP T (φ) = 1 +
OP T (φ)
2
2
OP T (ϕ) = OP T (φ) +
0
0
According to = 140
139 − ε for any > 0 on 3-bounded graphs, then the inapproximability factor
can be derived from inapproximability optimal substructure framework as
δ>
C
1
1
>1+
−ε=1+
−ε
(C − 1) + 1
140C
140(2ζ(β)3β − 1)
where the last step follows from d = 3.
Theorem 4.5. There is no 1 +
Set on power-law graphs.
1
390(2ζ(β)3β −1)
approximation algorithm for Minimum Dominating
9
Proof. In this proof, we construct the power-law graph G(α,β) based on cycle-based embedding
technique in Theorem 4.1 from d-bounded graph Gd . Let φ and ϕ be feasible solutions of MDS
~
on Gd and G(α,β) . The optimal MDS on OP T (φ), clique K2 , cycle, d-regular
cycle and ~κ-branch~
d-cycles are n/2, n/4 and n/3 respectively. Let φ and ϕ be feasible solutions of MDS on Gd and
β
G(α,β) . Then we have C = 1 + (ζ(β)d −1)(d+1)
similar as the proof in Theorem 4.4.
2
391
According to = 390
in 3-bounded graphs, then the inapproximability factor can be derived
from inapproximability optimal substructure framework as
δ >1+
−1
1
=1+
C
390(2ζ(β)3β − 1)
where the last step follows from d = 3.
Theorem 4.6. MVC is hard to be approximated within
log c
2 1−(2+oc (1)) log
log c
1+ 1
ζ(β)cβ +c β (c+1)
on power-law graphs
under unique games conjecture.
Proof. By constructing the power-law graph G(α,β) based on cycle-based embedding technique
~
in Theorem 4.1 from d-bounded graph Gd , The optimal MVC on clique K2 , cycle, d-regular
cycle
~
are half number
κ-branch-d-cycles are all vertices. Thus, we
of vertices while the optimal MVC on ~
1
ζ(β)dβ −1+d β
have C = 1 +
2
(d+1)
since
1
N −n−∆
(ζ(β)dβ − 1)n + n β d
OP T (ϕ) ≤ OP T (φ) +
+ ∆ ≤ OP T (φ) +
2
2
ζ(β)dβ − 1 +
= OP T (φ) +
d
n
1− 1
n β
(4.2)
2
ζ(β)dβ − 1 +
(4.1)
d
1− 1
β
(d+1)
(d + 1)
≤ OP T (φ) +
OP T (φ)
2


1
ζ(β)dβ − 1 + d β (d + 1)

 OP T (φ)
≤ 1+
2
(4.3)
(4.4)
where φ and ϕ be feasible solutions of MVC on Gd and G(α,β) , ∆ is the maximum degree in
~
G(α,β) . The inequality (4.1) holds since there are at most ∆ vertices in ~κ-branch-d-cycle,
i.e.
α/β
1/β
∆=e
≤ n d; (4.3) holds since there are at least d + 1 vertices in a d-bounded graph and the
optimal MVC in a d-bounded graph is at least n/(d + 1).
According to = 2−(2+od (1)) log log d/ log d, then the inapproximability factor can be derived
from inapproximability optimal substructure framework as
log log c
2
1
−
(2
+
o
(1))
c
log c
−1
δ >1+
≥1+ 1
C
ζ(β)cβ + c β (c + 1)
10
where c is the smallest d satisfying the condition in [15]. The last inequality holds since function
f (x) = (1 − (2 + ox (1)) log log x/ log x)/g(x)(x + 1) is monotonously decreasing when f (x) > 0 for
all x > 0 when g(x) is monotonously increasing.
Theorem 4.7. ρ-PDS is hard to be approximated into 2 − (2 + od (1)) logloglogd d on d-bounded graphs
under unique games conjecture.
Proof. In this proof, we show the gap-preserving from MVC on (d/ρ)-bounded graph G = (V, E)
v1
v2
to ρ-PDS on d-bounded graph G0 = (V 0 , E 0 ). w.l.o.g., we assume that d and d/ρ are integers. We
construct a graph G0 = (V 0 , E 0 ) by adding new vertices and edges to G as follows. For each edge
1 , . . . , v k where 1 ≤ k ≤ b1/ρc and ρ ≤ 1/2. Then we add 2k
(vi , vj ) ∈ E, create k new vertices vij
ij
v3
v4
l , v ) and (v l , v ) for all l ∈ [1, k] as shown in Figure 2. Clearly, G0 = (V 0 , E 0 ) is a
new edges (vij
i
ij j
d-bounded graph.
V1k2
...
V112
v2
v1
v1
v2
V1k4
...
V213
V114
v4
...
V2k3
v3
V314
...
v4
v3
(a) Instance G = (V, E)
V3k4
(b) Reduced Instance G0 = (V 0 , E 0 )
Figure 2: The Reduction from MVC to ρ-MDS
...
0
Let φ and ϕ be feasible solutions to MVC
V1k2 on G and G respectively. We claim that OP T (φ) =
OP T (ϕ).
On one hand, if S = {v1 , v2 , . . . , vj } ∈ V is the minimum vertex cover on G. Then {v1 , v2 , . . . , vj }
is a ρ-PDS on G0 because each vertex in V 112has ρ of all neighbors in MVC and every new vertex
in V 0 \ V has at least one of twovneighbors in MVC.
v2 Thus OP T (φ) ≥ OP T (ϕ). One the other
1
hand, we can prove that OP T (ϕ) does not contain new vertices, that is, V 0 \ V . Consider a vertex
l for all v ∈ N (v ) and all l ∈ [1, k] are not needed to
vi ∈ V , if vi ∈ OP T (ϕ), the new vertices vij
j
i
k ...
V213ρ ...
V2k3
be selected. If vi 6∈ OPVT14(ϕ), V
it114has to be dominated by
proportion
of its all neighbors. That is,
l have to be selected since every v l has to
for each edge (vi , vj ) incident to vi , either vj or all vij
ij
l are selected in OP T (ϕ) for some edge (v , v ), v is still
be either selected or dominated. If all vij
i j
j
not dominated by enough vertices if there are some more edges incident to vj and the number of
v4
v3
l k is great than 1, that
is, b1/ρc1 ≥ 1. In this case, vj will be selected to dominate all
vertices vij
34
l . Thus, OP T (ϕ) does not contain newVvertices.
vij
Since the vertices in V selected is a solution
to ρ-MDS, that is, for each vertex vi in graph G, vi will be selected or at least the number of
k
neighbors of vi will be selected. Therefore,V3the
vertices in OP T (ϕ) consist of a vertex cover in G.
4
Thus OP T (φ) ≤ OP T (ϕ). Then we show the completeness and soundness as follows.
...
• If OP T (φ) = m ⇒ OP T (ϕ) = m
11
log(d/2)
m ⇒ OP T (ϕ) > 2 − (2 + od (1)) logloglogd d m
• If OP T (φ) > 2 − (2 + od (1)) loglog(d/2)
log log d
log log(d/ρ)
m > 2 − (2 + od (1))
m
OP T (ϕ) > 2 − (2 + od (1))
log(d/ρ)
log d
since the function f (x) = 2 − log log x/ log x is monotonously increasing for any x > 0.
Theorem 4.8. ρ-PDS is hard to be approximated into 1+
under unique games conjecture.
log c
2 1−(2+oc (1)) log
log c
2+(ζ(β)cβ −1)(c+1)
on power-law graphs
Proof. By constructing the power-law graph G(α,β) based on cycle-based embedding technique
in Theorem 4.1 from d-bounded graph Gd , According to the optimal MVC on OP T (φ), clique K2 ,
β
~
~
from
cycle, d-regular
cycle and ~κ-branch-d-cycles,
we have C = 1 + (ζ(β)d −1)(d+1)
2
OP T (ϕ) = OP T (φ) + n1 /2 + f (ρ)n2 + g(ρ)n3
(ζ(β)dβ − 1)(d + 1)
N −n
≤ 1+
OP T (φ)
≤ OP T (φ) +
2
2
(
1
, ρ≤ 1
, g(ρ) = 13 for all ρ ≤ 12 and φ, ϕ be feasible solutions of MVC on
where f (ρ) = 41 1 3
1
,
<
ρ
≤
3
3
2
Gd and G(α,β) . n1 , n2 and n3 are correspondent to the number of vertices in cliques K2 , cycle,
~
~
d-regular
cycle and ~κ-branch-d-cycle.
According to = 2−(2+od (1)) log log d/ log d, then the inapproximability factor can be derived
from inapproximability optimal substructure framework as
log log c
2
1
−
(2
+
o
(1))
c
log c
−1
δ >1+
≥1+
β
C
2 + (ζ(β)c − 1)(c + 1)
where c is the smallest d satisfying the condition in [15]. The last inequality holds since function
f (x) = (1 − (2 + ox (1)) log log x/ log x)/g(x)(x + 1) is monotonously decreasing when f (x) > 0 for
all x > 0 when g(x) is monotonously increasing.
5. More Inapproximability Results on Simple Power-Law Graphs
5.1. General Graphic Embedding Technique
In this section, we introduce a general graphic embedding technique to embed a d bounded
graph into a simple power-law graph. Before presenting the embedding technique, we first show
that a graph can be constructed in polynomial time from a class of integer sequences.
Lemma 5.1. Given a sequence of integers D = hd1 , d2 , . . . , dn i which is non-increasing, continuous
and the number of elements is at least as twice as the largest element in D, i.e. n ≥ 2d1 , it is
possible to construct a simple graph G whose d-degree sequence is D in polynomial time O(n2 log n).
12
Algorithm 2: Graphic Sequence Construction Algorithm
Input : d-degree sequence D = hd1 , d2 , . . . , dn i where d1 ≥ d2 ≥ . . . ≥ dn
Output: Graph H
1 while D 6= ∅ do
2
Connect vertex of d1 to vertices of d2 , d3 , . . . , dd1 +1 ;
3
d1 ← 0;
4
for i = 2 to d1 + 1 do
5
di ← di − 1;
6
end
7
Sort D in non-increasing order;
8
Remove all zero elements in D;
9 end
Proof. Starting with a set of individual vertices S of degree 0 and |S| = n, we iteratively connect
vertices together to increase their degrees up to given degree sequence. In each step, the leftover
vertex of highest degree is connected to other vertices one by one in the decreasing order of their
degrees. Then the sequence D will be resorted and all zero elements will be removed. The algorithm
stops until D is empty. The whole algorithm is shown as follows (Algorithm 2).
After each while loop, the new degree sequence, called D0 , is still continuous and its number of
elements is at least as twice as its maximum element. To show this, we consider three cases: (1)
If the maximum degree in D0 remains the same, there are at least d1 + 2 vertices in D. Since D
is continuous, the number of elements in D is at least d1 + 2 + d1 − 1, that is, 2d1 + 1. Therefore,
the number of elements in D0 is 2d1 , i.e. n ≥ 2d1 still holds. (2) If the maximum degree in D0
is decreased by 1, there are at least 2 elements of degree d1 in D. Thus, at most one element in
D will become 0. Then we have n ≥ 2d1 − 2 = 2(d1 − 1). (3) If the maximum degree in D0 is
decreased by 2, there are at most two element in D becoming 0. Thus, n ≥ 2d1 − 3 > 2(d1 − 2).
The time complexity of the algorithm is O(n2 log n) since there are at most n iterations and
each iteration takes at most O(n log n) to sort the new sequence D.
Theorem 5.1 (Graphic Embedding Technique). Any d-bounded graph Gd can be embedded
into a simple power-law graph G(α,β) with β > 1 in polynomial time such that Gd is a maximal
component and the number of vertices in G(α,β) can be polynomially bounded by the number of
vertices in Gd .
Proof. Given a d-bounded degree graph Gd = (V, E) and β > 1, we construct a power-law graph
G(α,β) of exponential factor β which includes Gd as a set of maximal components. The construction
is shown as Algorithm 3.
Algorithm 3: Graphic Embedding Algorithm
1
2
3
β
α ← max{ β−1
(ln 4 + β ln d), ln 2 + ln n + β ln d} and corresponding G(α,β) ;
D be the d-degree sequence of G(α,β) \ Gd ;
Construct G(α,β) \ Gd using Algorithm 2.
According to the lemma 5.1, the above construction is valid and finishes in polynomial time.
13
Then we show that N is upper bounded by ζ(β)2dβ n, where n and N are the number of vertices
in Gd and Gα,β respectively. From the construction, we know either
α≥
α
β
eα
(ln 4 + β ln d) ⇒ α ≥ ln 4 + β ln d + α/β ⇒ β ≥ 4e β
β−1
d
or
α ≥ ln 2 + ln n + β ln d ⇒
eα
≥ 2n
dβ
α
α
α
Therefore, de β ≥ 2e β + n. Note that de β is the number of vertices of degree d. In addition,
G has at most n vertices of degree d, so D is continuous degree sequence and has the number of
vertices at least as twice as the maximum degree.
In addition, when n is large enough, we have α = ln 2 + ln n + β ln d. Hence, the number of
vertices N in Gα,β is bound as N ≤ ζ(β)eα = 2ζ(β)dβ n, i.e. the number of vertices of Gα,β is
polynomial bounded by the number of vertices in Gd .
5.2. Inapproximability of MIS, MVC and MDS
Theorem 5.2. For any ε > 0, it is NP-hard to approximate Maximum Independent Set within
1
1 + 1120ζ(β)3
β − ε on simple power-law graphs.
Proof. In this proof, we construct the simple power-law graph G(α,β) based on graphic embedding
technique in Theorem 5.1 from d-bounded graph Gd . Let φ and ϕ be feasible solutions of MIS on
Gd and G(α,β) . Since OP T (φ) ≥ n/(d + 1) on d-bounded graphs and N ≤ 2ζ(β)dβ n, we further
have C = 2ζ(β)dβ (d + 1) from
OP T (ϕ) ≤ N ≤ 2ζ(β)dβ n ≤ 2ζ(β)dβ (d + 1)OP T (φ)
0
0
According to = 140
139 − ε for any > 0 on 3-bounded graphs, then the inapproximability factor
can be derived from inapproximability optimal substructure framework as
δ>
1
1
C
=1+
−ε>1+
−ε
(C − 1) + 1
140C − 1
1120ζ(β)3β
1
3120ζ(β)3β
on
Proof. From the proof of Theorem 5.2, we have C = 2ζ(β)dβ (d + 1). Then according to =
on 3-bounded graphs, we have
391
390
Theorem 5.3. It is NP-hard to approximate Minimum Dominating Set within 1 +
power-law graphs.
δ >1+
−1
1
≥1+
C
3120ζ(β)3β
2−(2+oc (1)) log log c
log c
Theorem 5.4. There is no 1 +
approximation algorithm of Minimum Vertex
2ζ(β)cβ (c+1)
Cover on power-law graphs under unique games conjecture.
14
Proof. Similar as the proof of Theorem 5.3, we have C = 2ζ(β)dβ (d+1). Then according to = 2−
(2 + od (1)) log log d/ log d, then the inapproximability factor can be derived from inapproximability
optimal substructure framework as
log log c
2 − (2 + oc (1)) log c
−1
≥1+
C
2ζ(β)cβ (c + 1)
where c is the smallest d satisfying the condition in [15].
δ >1+
2−(2+oc (1)) log log c
log c
Theorem 5.5. There is no 1 + 2ζ(β)cβ (c+1)
Dominating Set on power-law graphs.
approximation algorithm for Minimum Positive
Proof. Similar as Theorem 5.5, the proof follows from Theorem 4.7.
5.3. Maximum Clique, Minimum Coloring
Lemma 5.2 (Ferrante et al. [1]). Let G = (V, E) be a simple graph with n vertices and β ≥ 1.
Let α ≥ max{4β, β log n + log(n + 1)}. Then, G2 = G \ G1 is a bipartite graph.
P
Lemma
5.3. Given a function f (x) (x ∈ Z, f (x) ∈ Z+ ) monotonously decreases, then x f (x) ≤
R
x f (x).
P α/β β
Corollary 5.1. eα ei=1 d1 < (eα − eα/β )/(β − 1).
Theorem 5.6. Maximum Clique cannot be approximated within O n1/(β+1)− on large power-law
graphs with β > 1 and n > 54 for any > 0 unless NP=ZPP.
Proof. In [1], the authors proved the hardness of Maximum Clique problem on power-law graphs.
Here we use the same construction. According to Lemma 5.2, G2 = G \ G1 is a bipartite graph
when α ≥ max{4β, β log n + log(n + 1)} for any β ≥ 1. Let φ be a solution on general graph G
and ϕ be a solution on power-law graph G2 . We show the completeness and soundness.
• If OP T (φ) = m ⇒ OP T (ϕ) = m
If OP T (φ) ≤ 2 on graph G, we can solve clique problem in polynomial time by iterating the
edges and their endpoints one by one. However, G is not a general graph in this case. w.l.o.g,
assuming OP T (φ) > 2, then OP T (ϕ) = OP T (φ) > 2 since the maximum clique on bipartite
graph is 2.
0
• If OP T (φ) ≤ m/n1− ⇒ OP T (ϕ) < O 1/(N 1/(β+1)− ) m
In this case, we consider the case that 4β < β log n + log(n + 1), that is, n > 54. According
to Lemma 5.2, let α = β log n + log(n + 1). From Corollary 5.1, we have
∆ β
X
1
eα − eα/β
nβ (n + 1) − n(n + 1)1/β
2nβ+1 − n
=
<
i
β−1
β−1
β−1
i=1
0
Therefore, OP T (ϕ) = OP T (φ) ≤ m/n1− < O m/ N 1/(β+1)− .
α
N =e
<
Corollary 5.2. Minimum Coloring problem cannot be approximated within O n1/(β+1)− on large
power-law graphs with β > 1 and n > 54 for any > 0 unless NP=ZPP.
15
6. Relationship between β and Approximation Hardness
As shown in previous sections, many hardness and inapproximability results are dependent on
β. In this section, we analyze the hardness of some optimal substructure problems based on β by
showing that trivial greedy algorithms can achieve constant guarantee factors for MIS and MDS.
Lemma 6.1. When β > 2, the size of MDS of a power-law graph is greater than Cn where n is
the number of vertices, C is some constant only dependent on β.
Proof. Let S = (v1 , v2 , . . . , vt ) of degrees d1 , d2 , . . . , dt be the MDS of power-law graph G(α,β) .
Observing that the total degrees of vertices P
in dominating set must be at least the number of
vertices outside the dominating set, we have i=t
i=1 di ≥ |V \ S|. With a given total degree, a set
of vertices has
size when it includes the vertices of highest degrees. Since the function
P minimum
1
ζ(β − 1) = ∞
converges
when β > 2, there exists a constant t0 = t0 (β) such that
i=1 iβ−1
α X
t0 α ∆
X
e
e
i β ≥
i
iβ
i=1
i=t0
where α is any large enough constant. Thus the size of MDS is at least
!
tX
∆ α
0 −1
X
e
1
≈ ζ(β) −
eα ≈ C|V |
iβ
iβ
i=t0
where C = (ζ(β) −
i=1
Pt0
1
i=1 iβ )/(ζ(β)).
Consider the greedy algorithm which selects from the vertices of the highest degree to the
lowest. In the worst case, it selects all vertices with degree greater than 1 and a half of vertices
with degree 1 to form a dominating set. The approximation factor of this simple algorithm is a
constant.
Corollary 6.1. Given a power-law graph with β > 2, the greedy algorithm
αselects
vertices
P∆that
β + 1 eα ≈
in decreasing order of degrees provides a dominating set of size at most
e
/i
i=2
2
P0
(ζ(β) − 1/2)eα . Thus the approximation ratio is (ζ(β) − 12 )/(ζ(β) − ti=1
1/iβ ).
Let us consider another maximization problem MIS, we propose a greedy algorithm Power-lawGreedy-MIS as follows. We sort the vertices in non-increasing order of degrees and start checking
from the vertex of lowest degree. If the vertex is not adjacent to any selected vertex, it is selected.
The set of selected vertices forms an independent set with the size at least a half the number of
vertices of degree 1 which is eα /2. The size of MIS is at most a half of number of vertices. Thus,
the following lemma holds.
Lemma 6.2. Power-law-Greedy-MIS has factor 1/(2ζ(β)) on power-law graphs with β > 1.
16
7. Conclusion
In this paper, we analyzed the approximation hardness and inapproximability of optimal substructure problems on power-law graphs. These problems are only illustrated not be able to
approximated into some constant factors on both general and simple power-law graphs although
they remain APX-hard. However, we also notice that the gap between inapproximability factor
and the simple constant approximation ratio of these problems is still not small enough and the
hardness on power-law graph is weaker than that on degree bounded graphs. A question is raised:
Is there any efficient reduction which is not from bounded graph will improve the hardness results
on power-law graphs? Can we get stronger hardness results based on some specific power-law
models? For example, if the number of vertices only follow power-law distribution when degree is
larger than some constant i0 , we can reduce from graph of degree bounded by i0 and get better
results.
On the contrary, we also show that Max Clique and Graph Coloring are still very hard
to be approximated since the optimal solutions to these problems are dependent on the structure
of local graph component rather than global graph. In other words, the power-law distribution
in degree sequence does not help much for such optimization problems without the property of
optimal substructure.
Acknowledgment
This work is partially supported by NSF Career Award ] 0953284, DTRA, Young Investigator
Award, Basic Research Program ] HDTRA1-09-1-0061 and DTRA ] HDTRA1-08-10.
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Appendix A. Embedding Construction with β < 1
Ferrante et. al. [1] proved the NP-hardness of MIS, MDS, and MVC where β < 1 based on
Lemma 7.1 which is invalid. A counter-example is as follows. Let D1 = h3, 2, 2, 1i and D2 =
h7, 6, 5, 4, 3, 2, 2, 1i then D1 is eligible and Y1 = h1, 2, 1i, Y2 = h1, 2, 1, 1, 1, 1, 1i but D2 is NOT
eligible with fD2 (4) < 0. In this appendix, we present an alternative lemma to prove the hardness
of these problems on power-law graphs with β < 1.
Definition 7.1 (d-Degree Sequence). Given a graph G = (V, E), the d-degree sequence of G is
a sequence D = hd1 , d2 , . . . , dn i of vertex degrees in non-increasing order.
Definition 7.2 (y-Degree Sequence). Given a graph G = (V, E), the y-degree sequence of G is
a sequence Y = hy1 , y2 , ..., ym i where m is the maximum degree of G and yi = |{u|u ∈ V ∧ deg(u) =
i}|.
Definition 7.3 (Eligible Sequences). A sequence of integers S = hs1 , . . . , sn i is eligible if s1 ≥
s2 ≥ . . . ≥ sn and fS (k) ≥ 0 for all k ∈ [n], where
fS (k) = k(k − 1) +
n
X
i=k+1
min{k, si } −
k
X
si
i=1
Lemma 7.1 (Invalid Lemma in [1]). Let Y1 and Y2 be two y-degree sequences with m1 and m2
elements respectively such that (1) Y1 (i) ≤ Y2 (i), ∀1 ≤ i ≤ m1 , and (2) two corresponding d-degree
sequences D1 and D2 are contiguous. If D1 is eligible then D2 is eligible.
Erdős and Gallai [23] showed that an integer sequence is graphic - d-degree sequence of an
graph, if and only if it is eligible and the total of all elements is even. Then Havel and Hakimi [24]
gave an algorithm to construct a simple graph from a degree sequence.
Lemma 7.2 ([24]). A sequence of integers D = hd1 , . . . , dn i is graphic if and only if it is nonincreasing and the sequence of values D0 = hd2 − 1, d3 − 1, . . . , dd1 +1 − 1, dd1 +2 , . . . , dn i sorted in
non-increasing order is graphic.
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We now prove the following lemma, which can substitute Lemma 7.1 for the NP-hardness proof
in [1].
Lemma 7.3. Given an undirected graph G = (V, E), 0 < β < 1, there exists polynomial time
algorithm to construct a power-law graph G0 = (V 0 , E 0 ) of exponential factor β such that G is a set
of maximal components of G0 .
Proof. To construct G0 , we choose α = max{β ln(n−1)+ln(n+2), 3 ln 2}. Then beα /((n−1)β )c >
n + 2, i.e. there are at least 2 vertices of degree d in G0 \ G if there are a least 2 vertices of degree d
in G0 . According to the definition, the total degrees of all vertices in G0 and G are even. Therefore,
the lemma will follow if we prove that the degree sequence D of G0 \ G is eligible.
In D, the maximum degree is beα/β c. There is only one vertex of degree i if 1 ≤ eα /iβ < 2, i.e.
α/β
e
≥ i > (eα /2)1/β .
Let us consider fD (k) in two cases:
1. Case 1: k ≤ eα/β /2
fD (k) = k(k − 1) +
n
X
min{k, di } −
> k(k − 1) +
i=k
k+
k−1
X
i=B
di
i=1
i=k+1
T
−k
X
k
X
i+
B−1
X
i=1
2−
k
X
(T − k + 1)
i=1
= k(T − k) + (k − B)(k − 1 + B)/2 + B(B − 1) − k(2T − k + 1)/2
= (B 2 − B)/2 − k
where T = eα/β and B = (eα /2)1/β + 1. Note that α/β > ln 2 (2/β + 1) since α > 3 ln 2
α 1/β α/β and 0 < β < 1. Hence (eα /2)1/β + 1
(e /2)
> e
≥ 2k, that is, fD (k) > 0.
2. Case 2: k > eα/β /2
j
k
fD (k + 1) ≥ fD (k) + 2k − 2dk+1 ≥ fD (k) ≥ . . . ≥ fD ( eα/β /2 ) > 0
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